• Home
  • Textbooks
  • Advanced Engineering Mathematics, International Student Edition
  • The Fourier Integral and Fourier Transforms

Advanced Engineering Mathematics, International Student Edition

Peter V. O'Neil

Chapter 15

The Fourier Integral and Fourier Transforms - all with Video Answers

Educators

Section 1

The Fourier Integral

Problem 1

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}x & \text { for }-\pi \leq x \leq \pi \\ 0 & \text { for }|x|>\pi\end{cases}$

Check back soon!

Problem 2

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}k & \text { for }-10 \leq x \leq 10 \\ 0 & \text { if }|x|>10\end{cases}$

Check back soon!

Problem 3

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)=\left\{\begin{aligned}-1 & \text { for }-\pi \leq x \leq 0 \\ 1 & \text { for } 0<x \leq \pi \\ 0 & \text { for }|x|>\pi\end{aligned}\right.$

Check back soon!

Problem 4

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}\sin (x) & \text { for }-4 \leq x \leq 0 \\ \cos (x) & \text { for } 0<x \leq 4 \\ 0 & \text { for }|x|>4\end{cases}$

Check back soon!

Problem 5

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}x^2 & \text { for }-100 \leq x \leq 100 \\ 0 & \text { for }|x|>100\end{cases}$

Check back soon!

Problem 6

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}|x| & \text { for }-\pi \leq x \leq 2 \pi \\ 0 & \text { for } x<-\pi \text { and for } x>2 \pi\end{cases}$

Check back soon!

Problem 7

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}\sin (x) & \text { for }-3 \pi \leq x \leq \pi \\ 0 & \text { for } x<-3 \pi \text { and for } x>\pi\end{cases}$

Check back soon!

Problem 8

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)= \begin{cases}1 / 2 & \text { for }-5 \leq x<1 \\ 1 & \text { for } 1 \leq x \leq 5 \\ 0 & \text { for }|x|>5\end{cases}$

Check back soon!

Problem 9

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)=e^{-|x|}$

Check back soon!

Problem 10

In each of Problems 1 through 10, expand the function in a Fourier integral and determine what this integral converges to.
$f(x)=x e^{-|4 x|}$

Check back soon!

Problem 11

Show that the Fourier integral of $f$ can be written
$$
\lim _{\omega \rightarrow \infty} \frac{1}{\pi} \int_{-\infty}^{\infty} f(t) \frac{\sin (\omega(t-x))}{t-x} d t
$$

Check back soon!